I've had my wii for around 3 years and during all that time, when I play a game, there are noises that come from the disc drive.
I've never taken it that seriously, but now I am kind of worried that my disc drive might be damaged and my games might be consequently damaged too.
Is the drive supposed to be silent or are the noises normal?

Depends on the game.
Each game is different, reading from disc as you play.
Some are reading more, some less.
And, yes, the noises are normal.
Remember the PS1 or the Dreamcast? Those consoles made lots of noise
also. The Wii is no different.
Put your fears to rest, young man.....